In the broad band network bolstering the internet industry, an optical communication is adopted. In transmission and reception of light in this optical communication, a laser using a compound semiconductor of a Group III-V, a Group II-VI or the like is used.
Among various structures proposed in the compound-semiconductor laser, a double hetero-structure is general. The double hetero-structure has a structure where two different kinds of compound semiconductors are used; a compound semiconductor having a smaller band gap is sandwiched by compound semiconductors each having a larger band gap. For the purpose of manufacturing the double hetero-structure, a compound semiconductor of conductivity which is an n-type, a compound semiconductor of conductivity which is i-type and is not doped, and a compound semiconductor of conductivity which is n-type are continuously epitaxially grown on a substrate to be laminated in a vertical direction. At the same time, it is necessary to pay attention to a band structure of the i-type compound semiconductor which is sandwiched in-between and not doped. It is important that a band gap of the i-type compound semiconductor is smaller than that of each of the n-type and p-type compound semiconductors, a conduction band level of the i-type is lower than a conduction band level of the n-type, and a valence band level of the i-type is higher than a valence band level of the p-type.
That is, both of electrons and holes are confined within the i-type region. This structure makes it easier for the electrons and holes to stay in a same region; therefore, the probability that the electrons and holes collide to annihilate each other increases, which improves a luminous efficiency as a result. In addition, a refractive index has a tendency to be larger as the band gap becomes smaller. Materials are, therefore, selected in such a way that the refractive index of the i-type compound semiconductor is larger than each of the refractive index of the n-type and p-type compound semiconductors, which results in that the light is also confined in the i-type compound semiconductor. The light thus confined efficiently induces a recombination of the electrons and holes which form an inverted distribution and leads to the laser oscillation.
Big amounts of long-distance information communication are made at once by the optical communication using the compound semiconductor which efficiently emits light in the manner as above. That is to say, the information processing and storage are executed on the LSI using silicon as a backbone, and the transmission of information is carried out by the laser using the compound semiconductor(s) as a backbone. If the silicon can be made to emit light with high efficiency, then, the electronic device and the light-emitting element can be integrated together with each other on a silicon chip. Therefore, its industrial value is enormous. Then, the research to make the silicon emit light has been largely done.
When the light-emitting element is to be manufactured on the silicon LSI to increase the integration degree, it will be effective as the second best to manufacture a highly efficient light-emitting element by germanium. This is because the degree of difficulty in manufacturing the light-emitting element by germanium is higher than that in an on-chip formation of the silicon light-emitting element, but is much lower than that in an on-chip formation of the compound semiconductor light-emitting element. In contrast, when a logic circuit, the light-emitting element, and an optical interconnection (optical waveguide) are manufactured on a same chip, a volume the waveguide occupies becomes larger than that the light-emitting element occupies. If the waveguide can be formed of silicon, the entire formation efficiency will improve. In this case, if the light-emitting element is also made of silicon a wavelength of the emitted light and a wavelength of light which tends to be absorbed in the waveguide will agree with each other. Hence, the germanium light-emitting element which emits light with a wavelength different from an absorption wavelength of silicon might be rather useful in some cases.
For the integration of the light-emitting elements, it is desirable to cause either silicon or germanium to efficiently emit light, which is not easy because germanium and silicon have an indirect transition band structure. The indirect transition band structure means a band structure in which a point k at which the energy of the conduction band becomes lowest (a position on a space k), and a point k at which the energy of the valence band becomes highest do not agree with each other. In the case of silicon, the highest energy point of the valence band is a point k called a point Γ of (kx, ky, kz)=(0, 0, 0). However, the lowest energy point of the conduction band is not present in the point Γ, but is present between the point Γ and a point X (±π/a, 0, 0), (0, ±π/a, 0), (0, 0, ±π/a). More specifically, letting a be a lattice constant with k0=0.85*π/a, the lowest energy point of the conduction band is present in a state of being regenerated to six points: (0, 0, ±k0), (0, ±k0, 0), (±k0, 0, 0). In the case of germanium, the highest energy point of the valence band is the point Γ. However, the lowest energy point of the valence band is present in a state of being regenerated to eight points (since the point L which is a point symmetry with the point Γ as a center is equivalent to one another, the number of independent ones is four): (±π/2a, ±π/2a, ±π/2a) which are called point L.
Whereas, many compound semiconductors are each called a direct transition semiconductor because the valence band highest energy point and the conduction band lowest energy point are each present at the point Γ.
A description will now be given with respect to the reason the luminous efficiency is lower in the indirect transition semiconductor, and the luminous efficiency is higher in the direct transition semiconductor. As has been described, to make the semiconductor element emit light, the electron and hole must collide to annihilate each other, and an energy difference between these two must be extracted as the light. In this case, the conservation law of energy and that of momentum must be both met. The electron has the energy level in the conduction band, and the hole has the energy level in a portion of valence band without the electron. The difference between these two turns into the energy the light possesses; extremely energy has different wavelength. The energy difference between the conduction band and the valence band, that is, the size of the band gap determines the wavelength of the light, that is, the color. It is thus impossible to find out an enormous difficulty in the establishment of the conservation law of energy.
On the other hand, since the phenomenon of the collision between the electron and hole is involved in the light emission, the momentum must be conserved. The momentum of the electron and hole is each proportional to a vector k representing the point k. According to the quantum mechanics as the law which governs the microscopic world, although the electron, the hole, and the photon (quantum of light) are each the wave, they are each scattered as the particle at the same time. Therefore, the law of conservation of momentum holds. The momentum is qualitatively a scale which quantifies how much momentum the particle is flicked by during the collision. It is understood from the dispersion relationship of the light (ω=ck where ω is the angular frequency, c is the light velocity, and k is the momentum of the photon) and the energy that the momentum of the photon within the crystal is estimated to be almost zero. This means that even if there is a phenomenon in which the light flicks the matter by collision, it hardly influences the matter to be scattered thereby, which agrees with our intuition.
Since in silicon and germanium, the energy minimum point (the valence band highest point) of the hole is present at the point Γ, the hole hardly has the momentum. However, since the energy minimum point (the conduction band energy lowest point) of the electron is present close to a point X in the case of silicon, and is present at the point Γ in the case of germanium, the electron has the large momentum.
Therefore, in silicon or germanium, in the process in which the electron and hole simply collide with each other, the conservation law of momentum, and the conservation law of energy cannot be met at the same time. Then, only a pair of electron and hole which have met the law of conservation of momentum and the conservation law of energy one way or another are converted into the light by, for example, absorbing or discharging the photon as the quantum of the lattice vibration in the crystal. Such a process can be physically present, but the probability of occurrence of the process is small because such process is a high-order scattering process such that the electron, the hole, and the photon collide with one another at a same time. Therefore, it is known that the luminous efficiency is extremely low in silicon or germanium as the indirect transition semiconductor.
By contrast, in most of direct transition compound semiconductors, the valence band highest energy and the conduction band lowest energy are each present at the point Γ. Therefore, the law of conservation of momentum, and the conservation law of energy can be both satisfied. Thus, the luminous efficiency is high in the compound semiconductor. A transistor laser element in which a laser using a compound semiconductor having the high luminous efficiency is driven by a bipolar transistor made of a compound semiconductor is reported in the following non-patent document 1.
It is known that although the luminous efficiency is extremely low in silicon in a bulk state as described, the luminous efficiency of silicon can be enhanced when in a porous state or a nanoparticle state. For example, silicon which has been anodized in a hydrofluoric acid solution is in the porous state, thereby emitting light at a room temperature and in a visible light wavelength band in the following non-patent document 2. Although the mechanics concerned is not perfectly elucidated, it is considered that a quantum size effect which is caused due to presence of silicon confined in a narrow region by formation of a porous silicon oxide is important. It is considered that since in silicon having the small size, the positions of the electrons is confined within the region concerned, and conversely, the momentum is not determined by the uncertainty principle of the quantum mechanics, thus making the recombination of the electron and hole easy to occur.
As another method using silicon, for example, it is described in the following non-patent document 3 that Er ions were implanted into a pn junction formed in a Si substrate, thereby making it possible to create a light-emitting diode (LED) becoming a light-emitting element. When the Er ions are implanted into the Si substrate, Er ions form impurity levels which are spatially localized. The momentum of the trapped electrons become effectively zero when the electrons present in the conduction band of Si are trapped in the impurity levels which the Er ions form. Thus, the electrons are able to be recombined with the holes in the valence band, thereby emitting the light. Since the light emission through the Er ions has a wavelength of 1.54 μm, the light could be propagated without being absorbed in surrounding silicon. In addition, this wavelength loses little when an existing optical fiber is used. Therefore, even when the Si-based LED using the Er ions is in practical use by the future technical innovation, the existing optical fiber network can be utilized. The Si-based LED is expected not to require a large scale equipment investment.
Moreover, as another method using silicon, for example, it is described in the following non-patent documents (4, 5) that Er ions were implanted into a silicon nanoparticle based on a combination of the quantum size effect described above, and an idea of the ER ion, whereby an efficiency was enhanced, thereby making it possible to carry out the light emission.
In the prior art described above, it was thought that for the purpose of making silicon emit light, to move the momentum away from a point of k0 through the uncertainty principle by changing the band structure of the conduction band of silicon to the band structure of the bulk, it was only necessary to make silicon become a porous state, a nanoparticle state or the like by the quantum size effect. However, there is a problem that if, for example, silicon having such a structure as the nanoparticle is formed, then, a surface of the silicon nanoparticle is oxidized to form a silicon dioxide on the surface due to the feature that a silicon surface is extremely easy to oxidize. Since the silicon dioxide is an insulator which is extremely large in band gap, there is caused a problem that if the silicon dioxide is formed on the surface, then, the electrons or the holes cannot be efficiently implanted. Therefore, in the conventional light-emitting element, there is a problem that even if a high intensity is obtained in photoluminescence, the efficiency is extremely reduced in electroluminescence. In addition, during the light emission, the crystallinity of the matter becoming the light-emitting layer becomes important. However, in the nanoparticle formed by a Chemical Vapor Deposition (CVD) or in the structure in which a large number of irregular holes are opened by the anodic oxidation, there is a problem that its crystallinity is lower compared to the single crystal. With low crystallinity, the light emission through defect levels is generated. However, in the light emission utilizing the defects, there is a problem that with low crystallinity it is impossible to manufacture the element which withstands actual use such as the information communication.
As has been described, although the effort for making silicon emit light has been made by the various techniques such as porous silicon, the nanoparticle, and the Er doping, the luminous efficiency has not so high as to reach the practical level.
We reported that a light-emitting element includes a first electrode portion into which electrons are implanted, a second electrode portion into which holes are implanted, and an a light-emitting portion electrically connected to the first electrode portion and the second electrode portion; the light-emitting portion is made of a single crystal silicon; and the light-emitting portion has a first surface (upper surface) and a second surface (lower surface) facing the first surface. Also, in this case, we reported that a plain orientation of each of the first and second surfaces is made a (001) plane, and a thickness of the light-emitting portion in a direction perpendicular to the first and second surfaces is thinned, thereby obtaining the light-emitting element which can be readily formed on a substrate such as silicon by using a normal silicon process, and which emits light at a high efficiency (the following PTL 1). For this, it is possible to understand that when the electrons are confined in the extremely narrow region represented by a extremely thin single crystal silicon film or the like, in the electron state of the bulk, even in the case of the matter like silicon in which the electrons in the conduction band are absent in the point Γ, the electrons cannot carry out motion effectively in the vertical direction of the thin film. This shows qualitatively a matter of cause that since the direction vertical to the thin film becomes absent, the electrons cannot be moved in the direction vertical to the thin film. That is to say, it is thought that even in the case of an ultra-decrease in film of silicon, in the bulk, the indirect transition semiconductor effectively turns into the direct transition semiconductor by the quantum confinement effect (two-dimensional confinement effect). Hereinafter, the principles of the light emission and the demonstration result thereof will be shown.
A description will now be given with respect to the principles for making a Group IV semiconductor such as silicon and germanium following the same efficiently emit light. A wave function ψ(r) representing the state of the electrons in the crystal such as silicon can be expressed like (Expression 1) in extremely high approximation.
[Expression 1]ψ(r)=φk0(r)ξ(r)  (1)
Here, k0 is a momentum giving a valley of a band of a conduction band, r=(x, y, z) represents a position on a space, Φk0(r) gives a Bloch function in the valley of the band of the conduction band, and ξ(r) represents an envelope function, Φk0(r) is expressed as (Expression 2) by using a periodic function uk0(r+a)=uk0(r) reflecting the periodicity for a unit lattice vector “a” in a crystal.
[Expression 2]φk0(r)=uk0(r)eik0r  (2)
As apparent from this as well, the wave function ψ(r) violently vibrates as a function of a distance of an atomic-scale distance. In contrast, the envelope function ξ(r) represents a component which exhibits a gentle change in the atomic scale, and represents a response to the physical shape of the semiconductor and the external field which is applied from the circumferences. When a thought is made including a case where ψ(r) is the wave function in the semiconductor structure which is not necessarily the bulk crystal, but has the finite size, Expression which ξ(r) should meet can be derived as Expression (3).
[Expression 3][ε(k0−i∇)+V(r)]ξ(r)=Eξ(r)  (3)
Here, ε=ε(k) represents a band structure in bulk of a conduction band electrode having a movement k, and a term which is obtained by substituting a sum of a differential vector −i∇, and a momentum k0 into the momentum k is represented in the form of ε(k0−i∇). In addition, V=V(r) is a potential which the electron feels. For example, when an insulator or another kind of semiconductor contacts a boundary portion of a semiconductor, V=V(r) gives a potential barrier. Also, an electric field is applied from the outside by an electric field effect, thereby making it also possible to adjust a value of V=V(r). For the sake of the simplicity here, attention is paid only to the change for the z direction of V.
In order to facilitate the understanding, specifically, supposing silicon on a (001) plane as the semiconductor as the semiconductor, as has been described, in the bulk, the valley of the conduction band existing in (0, 0, ±k0) in a kz direction can be approximated as (Expression 4).
                    [                  Expression          ⁢                                          ⁢          4                ]                                                                      ɛ          ⁡                      (            k            )                          =                                                            ℏ                2                                            2                ⁢                                  m                  t                  *                                                      ⁢                          (                                                k                  x                  2                                +                                  k                  y                  2                                            )                                +                                                    ℏ                2                                            2                ⁢                                  m                  l                  *                                                      ⁢                                          (                                                      k                    z                                    ∓                                      k                    0                                                  )                            2                                                          (        4        )            
Here, m*t and m*i represent effective masses in a silicon crystal which are respectively obtained from curvatures in short axis and long axis directions of the valley of the conduction band having a spheroidal shape. Then, Expression (3) is expressed as Expression (5).
                    [                  Expression          ⁢                                          ⁢          5                ]                                                                                  [                                                            -                                                            ℏ                      2                                                              2                      ⁢                                              m                        t                        *                                                                                            ⁢                                  (                                                            ∂                      x                      2                                        ⁢                                          +                                              ∂                        y                        2                                                                              )                                            -                                                                    ℏ                    2                                                        2                    ⁢                                          m                      l                      *                                                                      ⁢                                                      ∂                    z                    2                                    ⁢                                      +                                          V                      ⁡                                              (                        r                        )                                                                                                                  ]                    ⁢                      ξ            ⁡                          (              r              )                                      =                  E          ⁢                                          ⁢                      ξ            ⁡                          (              r              )                                                          (        5        )            
It is noted that letting (x, y) be the direction parallel with the (001) plane, w be a width, L be a length, and setting an envelope function as (Expression 6), (Expression 5) is simplified to (Expression 7).
                    [                  Expression          ⁢                                          ⁢          6                ]                                                                      ξ          ⁡                      (            r            )                          =                                            ⅇ                              ⅈ                ⁡                                  (                                                                                    k                        x                                            ⁢                      x                                        +                                                                  k                        x                                            ⁢                      y                                                        )                                                                                    L                ⁢                                                                  ⁢                W                                              ⁢                      χ            ⁡                          (              z              )                                                          (        6        )                                [                  Expression          ⁢                                          ⁢          7                ]                                                                                  [                                          -                                                      ℏ                    2                                                        2                    ⁢                                          m                      l                      *                                                                                  ⁢                                                ∂                  z                  2                                ⁢                                  +                                      V                    ⁡                                          (                      z                      )                                                                                            ]                    ⁢                      χ            ⁡                          (              z              )                                      =                  Δ          ⁢                                          ⁢          E          ⁢                                          ⁢                      χ            ⁡                          (              z              )                                                          (        7        )            
Here, ΔE represents an energy in the z direction and an entire energy of the electron measured from a bottom of the conduction band is expressed by (Expression 8).
                    [                  Expression          ⁢                                          ⁢          8                ]                                                            E        =                                                            ℏ                2                            ⁢                              k                x                2                                                    2              ⁢                              m                t                *                                              +                                                    ℏ                2                            ⁢                              k                y                2                                                    2              ⁢                              m                t                *                                              +                      Δ            ⁢                                                  ⁢            E                                              (        8        )            
Firstly, it should be confirmed that (Expression 7) reproduces the electron state of the bulk. To that end, it is only necessary to obtain a solution of a continuous state when V(r)=0 is set. This is confirmed from that when letting t be a thickness of the z direction, the envelope wave function becomes (Expression 9), and ΔE becomes (Expression 10).
                    [                  Expression          ⁢                                          ⁢          9                ]                                                                      χ          ⁡                      (            z            )                          =                              1                          t                                ⁢                      ⅇ                          ⅈ              ⁢                                                          ⁢                              k                z                            ⁢              z                                                          (        9        )                                [                  Expression          ⁢                                          ⁢          10                ]                                                                      Δ          ⁢                                          ⁢          E                =                                                            ℏ                2                            ⁡                              (                                                      k                    z                                    ∓                                      k                    0                                                  )                                      2                                2            ⁢                          m              l              *                                                          (        10        )            
That is to say, the wave function violently vibrates in a state in which it continuously spreads over the entire bulk crystal. At this time, it goes without saying that when letting kz be a momentum operator in the z direction, a quantum-mechanical expected value of the momentum in the z direction becomes (Expression 11).
                    [                  Expression          ⁢                                          ⁢          11                ]                                                                                                                〈                                                      k                    ^                                    z                                〉                            =                            ⁢                              ∫                                                                            ⅆ                      3                                        ⁢                    r                                    ⁢                                                                          ⁢                                                            ψ                      *                                        ⁡                                          (                      r                      )                                                        ⁢                                      (                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                              ∂                        z                                                              )                                    ⁢                                      ψ                    ⁡                                          (                      r                      )                                                                                                                                              =                            ⁢                                                k                  z                                ±                                  k                  0                                                                                        (        11        )            
That is to say, it is shown from the mathematical expression as well that in the indirect transition semiconductor such as silicon, since the probability that many electrons are present in points far away from the point Γ in the momentum space is predominantly high, they move with having greatly large momentums. The present invention uses the fact that in the case of the ultra-thin film in which t as the thickness in the z direction is greatly small, in the bulk, the indirect transition semiconductor effectively turns into the direct transition by the quantum confinement effect as the basic principles. Hereinafter, this respect will be described in detail.
To describe a story in a simplified manner, continuously, it is supposed that in a state of giving silicon as an example, silicon in which a thickness t in the z direction is greatly small vertically, adjacently contacts an insulator such as SiO2 having a large band gap or the vacuum or the atmosphere having a larger energy gap vertically in the z direction. As a system for which the same effect can be expected, for example, if the electrons are confined within the narrow region by the electric field effect or the like, then, the same effect can be expected. In these cases, the wave function of the electron in silicon becomes zero in a vertical interface in the z direction. Of course, strictly, the ooze of the quantum-mechanical wave function is preset. However, since the energy barrier is large, the ooze is exponentially reduced to the distance in the z direction. Hence, the approximation that the wave function becomes zero in the interface is approximately strictly true. Therefore, even if the potential V(r) which is applied from the outside is zero, then, the situation of the envelope wave function is different from the case where t is thick at all. Actually, the envelope wave function of the electrons and holes which are confined within such a quantum well are solved as (Expression 12) in the case of an even number: n=0, 2, 4, . . . where n is an index representing a discrete energy level, and become (Expression 13) in the case of an odd number: n=1, 3, 5, . . . . Thus, a value of the energy level can be expressed as (Expression 14) irrespective of whether n is an even number or an odd number.
                    [                  Expression          ⁢                                          ⁢          12                ]                                                                                  χ            n                    ⁡                      (            z            )                          =                                            2              t                                ⁢                      cos            ⁡                          (                              π                ⁢                                                                  ⁢                                  z                  t                                ⁢                                  (                                      n                    +                    1                                    )                                            )                                                          (        12        )                                [                  Expression          ⁢                                          ⁢          13                ]                                                                                  χ            n                    ⁡                      (            z            )                          =                                            2              t                                ⁢                      sin            ⁡                          (                              π                ⁢                                                                  ⁢                                  z                  t                                ⁢                                  (                                      n                    +                    1                                    )                                            )                                                          (        13        )                                [                  Expression          ⁢                                          ⁢          14                ]                                                                      Δ          ⁢                                          ⁢          E                =                                            ℏ              2                                      2              ⁢                              m                l                *                                              ⁢                                    π              2                                      t              2                                ⁢                                    (                              n                +                1                            )                        2                                              (        14        )            
It goes without saying that the state in which the energy is lowest is n=0. In showing the envelope wave function, it was supposed that the origin of the z-axis was set at the center of the thin film silicon, and the interface having the high energy barrier was present in z=±t/2. Here, a description will be given with respect to the property of the envelope wave function χn(z). When n is either 0 or an even number, the envelope wave function is symmetrical with respect to a sign change of z, and has the property of χn(z)=χn(−z). This is the as that parity is even. On the other hand, when n is an odd number, the envelope wave function has the property of χn(z)=−χn(−z). This is the as that parity is odd.
Since the envelope wave function has such a structure that the symmetry is reflected, when the contribution to the momentum by the envelope wave function is evaluated, (Expression 15) is obtained.
                    [                  Expression          ⁢                                          ⁢          15                ]                                                                                                                〈                                                      χ                    n                                    ⁢                                                                                                        k                        ^                                            z                                                                            ⁢                                      χ                    n                                                  〉                            =                            ⁢                              ∫                                                      ⅆ                    z                                    ⁢                                                                          ⁢                                                            χ                      n                      *                                        ⁡                                          (                      z                      )                                                        ⁢                                      (                                                                  -                        ⅈ                                            ⁢                                              ∂                        z                                                              )                                    ⁢                                                            χ                      n                                        ⁡                                          (                      z                      )                                                                                                                                              =                            ⁢              0                                                          (        15        )            
This has the extremely common property that since when χn(z) is differentiated with the z direction, the parity is changed to the parity which χn(z) originally has, when χn(z) is integrated with the z direction, it becomes zero. In a word, it is understood that there is the property in which since the electron is strongly restricted in the z direction, the envelope wave function becomes a standing wave, and thus the electron stops moving. This completely contracts with that the envelope wave function in the bulk state has an exponential form as given by (Expression 9), and the electron moves around the entire bulk crystal with the momentum. However, the full wave function for which up to the presence of the Bloch function is taken into consideration is obtained by substituting (Expression 2), (Expression 6), and (Expression 13) or (Expression 14) into the (Expression 1). Therefore, it is necessary to pay attention to that the quantum-mechanical expected value of the momentum in the z direction becomes (Expression 16).
                    [                  Expression          ⁢                                          ⁢          16                ]                                                                                                                〈                                                      k                    ^                                    z                                〉                            =                            ⁢                              ∫                                                                            ⅆ                      3                                        ⁢                    r                                    ⁢                                                                          ⁢                                                            ψ                      *                                        ⁡                                          (                      r                      )                                                        ⁢                                      (                                                                  -                        ⅈ                                            ⁢                                              ∂                        z                                                              )                                    ⁢                                      ψ                    ⁡                                          (                      r                      )                                                                                                                                              =                            ⁢                              ±                                  k                  0                                                                                        (        16        )            
In a word, when the envelope wave function is the bulk as the original property of the semiconductor material, since the valley of the conduction band is not present in the point Γ, but the valley is present at (0, 0, ±k0), the overall wave function reflects the property thereof. When the property is viewed in such a manner, even in the thin film the electron seems to be moving around with having the momentum ±k0, yet we take note that attention needs to be paid to this. In a word, for example, in the matter having the reversal symmetry in terms of the crystal like silicon, attention needs to be paid to that the valley (0, 0, +k0) and the valley of (0, 0, −k0) are equal in energy to each other, and thus the regeneration is caused. When the quantum-mechanical states having the energy levels which are extremely generally regenerated are confined within the spatially same region, hybrid is generated among these states. In a word, if the energy band which couples the valley (0, 0, +k0) and the valley of (0, 0, −k0) is extremely slightly present, two discrete levels form the bonding orbital and the anti-bonding orbital. For example, it would be thought that the coulomb interaction or the like between the electrons, which is not sufficiently contained in the band calculation, strongly acts between the electrons confined within the narrow region. The interaction acting between the electrons is called the electron correlation, and becomes a large problem in many transition metallic oxides including the high-temperature super-conduction. However, the interaction has not been the large problem in the bulk silicon until now by reflecting that the sp orbital in the original silicon atom has the large orbital. However, when the electrons are confined in such an extremely narrow region that the quantum-mechanical effect becomes important, since the coulomb interaction strongly acts on such a case, it will be impossible to disregard such a coulomb interaction between the electrons. If the coulomb interaction is taken in the proper way, and the matrix elements of Hamiltonian are calculated, then, the hybrid which couples the valley (0, 0, +k0) and the valley of (0, 0, −k0) is present therein. Also, if the Hamiltonian is diagonalized, then, it is understood that the Hamiltonian is divided into the bonding orbital and the anti-bonding orbital. This is similar to a process in which a hydrogen molecule is formed when two hydrogen atoms are brought close to each other. A method of evaluating such a system has been understood for about 70 years when the quantum mechanics was formed by Heitler-London. We firstly took note that the formation of the bonding state understood by Heitler-London is important in the coupling as well of the valleys even in the case where a Group IV semiconductor such as silicon is confined within the narrow region. In addition, even if such energetical bonding is not present at all, the standing wave which is not moving in the z-axis direction can be structured from unitary transformation of two states. This will now be described in more detail. The Bloch state has the property of u−k0(r)=uk0(r) from the reversal symmetry which the crystal has. Therefore, the Bloch wave functions of the valley (0, 0, +k0) and the valley of (0, 0, −k0) are expressed in the form of Φk0(r)=uk0(r)eik0z, and Φ−k0(r)=uk0(r)e−ik0z, respectively. Therefore, it is understood that it is only necessary to pay attention to a portion of e±k0z. In order to structure the new ground state from the sum of and difference between those wave functions, it is only necessary to make the transformation to (Expression 17) by the unitary transformation.
                    [                  Expression          ⁢                                          ⁢          17                ]                                                                                                                U                (                                                                                                    ⅇ                                                  ⅈ                          ⁢                                                                                                          ⁢                                                      k                            0                                                    ⁢                          z                                                                                                                                                                        ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                                                      k                            0                                                    ⁢                          z                                                                                                                    )                            =                            ⁢                                                1                                      2                                                  ⁢                                  (                                                                                    1                                                                    1                                                                                                                                      -                          ⅈ                                                                                            ⅈ                                                                              )                                ⁢                                  (                                                                                                              ⅇ                                                      ⅈ                            ⁢                                                                                                                  ⁢                                                          k                              0                                                        ⁢                            z                                                                                                                                                                                        ⅇ                                                                                    -                              ⅈ                                                        ⁢                                                                                                                  ⁢                                                          k                              0                                                        ⁢                            z                                                                                                                                )                                                                                                        =                            ⁢                                                2                                ⁢                                  (                                                                                                              cos                          ⁡                                                      (                                                                                          k                                0                                                            ⁢                              z                                                        )                                                                                                                                                                                        sin                          ⁡                                                      (                                                                                          k                                0                                                            ⁢                              z                                                        )                                                                                                                                )                                                                                        (        5        )            
Therefore, it is understood that the change in the wave function at the atomic level can be described by the wave function of two standing waves of 21/2uk0(r)cos(k0z) and 21/2uk0(r)sin(k0z). Also, when the entire wave functions are shown, they can be expressed by (Expression 18) and (Expression 19), respectively.
                    [                  Expression          ⁢                                          ⁢          18                ]                                                                      ψ          ⁡                      (            r            )                          =                              2                    ⁢                                    u                              k                0                                      ⁡                          (              r              )                                ⁢                      cos            ⁡                          (                                                k                  0                                ⁢                z                            )                                ⁢                      ξ            ⁡                          (              z              )                                                          (        18        )                                [                  Expression          ⁢                                          ⁢          19                ]                                                                      ψ          ⁡                      (            r            )                          =                              2                    ⁢                                    u                              k                0                                      ⁡                          (              r              )                                ⁢                      sin            ⁡                          (                                                k                  0                                ⁢                z                            )                                ⁢                      ξ            ⁡                          (              z              )                                                          (        19        )            
The expected value in the z-axis direction of the momentum in the state of either (Expression 18) or (Expression 19) becomes (Expression 20) by reflecting that the wave function expected value is the standing wave.
                    [                  Expression          ⁢                                          ⁢          20                ]                                                                                                                〈                                                      k                    ^                                    z                                〉                            =                            ⁢                              ∫                                                      ⅆ                    z                                    ⁢                                                                          ⁢                                                            ψ                      *                                        ⁡                                          (                      z                      )                                                        ⁢                                      (                                                                  -                        ⅈ                                            ⁢                                              ∂                        z                                                              )                                    ⁢                                      ψ                    ⁡                                          (                      z                      )                                                                                                                                              =                            ⁢              0                                                          (        20        )            
In a word, it is understood that the electron is not moving in the z direction at all. Since misunderstanding may occur in that only by changing the ground, the expected value of the momentum appears to be changing, and thus attention is paid thereto. In reality, the ground wave functions such as (Expression 18) and (Expression 19) are not eigenstate of the momentum. That is to say, the matrix element of the momentum operator becomes (Expression 21) when (Expression 18) and (Expression 19) are used, Diagonal matrix elements each becomes zero, and thus off-diagonal matrix elements become pure impurity members.
                    [                  Expression          ⁢                                          ⁢          21                ]                                                                                  U            ⁡                          (                                                                                          k                      0                                                                            0                                                                                        0                                                                              -                                              k                        0                                                                                                        )                                ⁢                      U                          -              1                                      =                  (                                                    0                                                              ⅈ                  ⁢                                                                          ⁢                                      k                    0                                                                                                                                            -                    ⅈ                                    ⁢                                                                          ⁢                                      k                    0                                                                              0                                              )                                    (        21        )            
Whether taking such a ground is physically suitable depends on the property of the system as an object. We assume the ultra-thin single crystal silicon film; however, since in such a case, the translation symmetry for the z-axis direction is crumbling, it is suitable to use √2uk0(r)cos(k0z) or √2uk0(r) sin(k0z) becoming the standing wave rather than to use uk0(r)e±ik0z as the eigenstate of the momentum. Contrary to this, when the bulk state is handled, since the translation symmetry is present, it is better to use uk0(r)e±ik0z. In addition, in the bulk state, the electron having the momentum of ±k0 violently moves around within the crystal. In this case, the electron is strongly scattered by the phonon or the like that is the quantum of the lattice vibration in the crystal, and the phase of the wave function has changed dynamically. Therefore, it is impossible to expect that a state in which the state of the momentum of +k0, and the state of the momentum of −k0 are coherently coupled is formed. By contrast to this, when the ultra-thin single crystal silicon film or the like, the electron is confined in such an extremely narrow region as to be thinner than a mean free path I as a length characterizing the scattering, even at the room temperature, the wave function sufficiently can form the standing wave in which the phase is determined. Qualitatively, this means that while the wave of the electron moves in and out the narrow region, the wave concerned becomes a steady wave which perfectly fits the size of that region.
As given a detailed description above by using the simple mathematical expressions, it is understood that when the electron is confined in the extremely narrow region represented by the ultra-thin single crystal silicon film or the like, in the electron state in the bulk, even in the case of the matter like silicon in which the electron in the conduction band is not present at the point Γ, the electron does not effectively move in the direction vertical to the thin film. Qualitatively, this shows a matter of course in which the electron cannot move in the direction vertical to the thin film since the direction vertical to the thin film is absent. In a word, this means that even if the electron moves in the crystal at the high speed in the bulk, since the direction in which the electron should move becomes absent in the first place in the thin film, the electron is compelled to stop.
Since the movement in the z-axis direction comes to be unable to be carried out, the band structure of the bulk is projected onto the plane of kz=0 to become a two-dimensional band structure. The system which is confined in the two-dimension in such a manner is called a two-dimensional electron system. In addition, if the thin film is not formed, but the ultra-fine line structure like a nanowire is formed, then, the dimension is further reduced, thereby making it possible to form one-dimensional electron system as well. In the thin film, the conduction band lowest energy point of (0, 0, ±k0) in the three-dimensional band structure of the bulk is projected onto the point Γ in the two-dimensional band structure.
We have reached an idea that the electron present at the point Γ of the two-dimensional band structure is efficiently recombined with the hole, and is ought to be used as the light-emitting element. In a word, the electron is confined, whereby the electron cannot freely move. Therefore, when the electron collides with the hole which has the small momentum because it is similarly present at the point Γ, the light which also has the small momentum can be discharged without breaking the law of conservation of the momentum and the law of conservation of energy. As described above, the momentum is the scale showing how much impact the particle is scattered when the particle collides with another particle. We took note that if the electron is confined in the narrow region to be unable to move, then, the momentum of the electron is lost. If the momentum of the electron becomes small, then, the law of conversation of momentum during the scattering which has been difficult with the conventional method comes to be able to hold. Therefore, even the Group IV semiconductor such as silicon efficiently emits light.
In the examples so far, in the case of silicon, since the conduction band lowest point in the vicinity of the point X in the bulk is projected onto the two-dimensional point Γ, the thin film needs to take such a crystal orientation that the (001) plane becomes the surface of the film. In the case of germanium, since the conduction band lowest point is present at the point Γ, in order to obtain the same effect, a (111) plane needs to become the surface of the film.
However, although germanium is the same indirect transition semiconductor as silicon, germanium is different from silicon in that not only the energy of the conduction band becomes lowest at the point L, but also the minimum point of the energy is present at the point Γ. Since an energy difference between the point Γ and the point L is about 0.14 eV, if a large amount of electrons are implanted, then, after the point k in the vicinity of the point L is met, the electrons can also be implanted into the point Γ. For this reason, in the light emission by the direct transition in the germanium thin film, there are two cases. In the first case, the electron at the conduction band lowest point which is derived from the point L and which is projected onto the point Γ gets involved in the light emission. In the second case, the electron at the minimum point of the conduction band which has been originally present from the phase of the bulk gets involved in the light emission. With regard to the matrix elements of the optical transition with the valence band maximum point, the conduction band minimum point derived from the point Γ is much larger than the conduction band lowest point derived from the point L. Therefore, if the electron implantation can be sufficiently carried out, then, the luminous efficiency is better in the case where the minimum point derived from the point Γ is utilized than in the other cases. However, since a necessary amount of implanted electrons is not small, this is not simple. Accordingly, which to positively use the lowest point derived from the point L or the minimum point derived from the point Γ to easily increase the luminous efficiency depends on the thickness of the thin film, the structure of the interface, the structure of the device, and the like, depending on the situation.
The PTL 2 based on the element structure of PTL 1 also shows the element structure in which the silicon thin films are arranged in fins at intervals each of which is at half the light emission wavelength λ, and which are combined with the waveguide is shown for the purpose of efficiently utilizing the silicon thin film as the light source of the silicon laser. Two methods are shown for manufacturing the fin-like structure. In the first method, after a fine line pattern is drawn on a mask, etching is deeply carried out. In the second method, after a silicon layer and a silicon-germanium layer are alternately laminated by using an MBE method, a silicon-germanium layer is selectively etched away, thereby leaving only a fin-like silicon layer.
It is shown in PTL 3 that a method of carrying out compression in a (001) direction of a silicon crystal, that is, in a direction vertical to a surface of a thin film is effective as a method of enhancing a luminous efficiency in the structure of PTLs (1, 2). When such compression that a volume is contracted by 2% is carried out, a pressure is about 1.5 Gp, and the luminous efficiency increases by from about 10% to about 30%. The reason why there is a width in the increase in the luminous efficiency is because the degree of the change differs depending on the thickness (the number of atomic layers). That effect is effective in the silicon thin film as well having a film thickness, which is capable of being readily formed, of about 30 atomic layers. Therefore, this effect is easy to utilize in enhancement of the efficiency of the light emitting element based on the silicon (001) thin film.
A description will now be given with respect to the double hetero-structure of the semiconductor laser which is previously realized in the general Group III-V compound semiconductor. Even in the Group III-V semiconductor which is relatively high in luminous efficiency, only by connecting a positive electrode and a negative electrode with the material left as is to implant the electrons and holes from the both sides, the electron and hole which did not collide with each other, did not emit light, and did not disappear reach the opposite-side electrode with the electron and hole passing each other without emitting the light to be absorbed therein, and thus the efficiency is low. Then, regions each having a large band gap than that of a light-emitting region are formed on both sides (on a positive side and a negative side) of the light-emitting region by using a technique such as epitaxial growth. As a result, this provides such a structure as to be an energy barrier for the holes and electrons. A semiconductor a band gap of which is originally larger than that of a semiconductor of the light-emitting region is used in the portion to be the barrier layer. This is called the double hetero-structure. The electron an the hole which have been implanted into the light-emitting region cannot go beyond the energy barrier by the double hetero-structure to be confined within the light-emitting region. Thus, the probability that the light is emitted increases because in the interim, the electron and hole are recombined with each other.